With mosquitoes plaguing their preparations, lightning flashing over the match, and an early goal disallowed, the Germans must have wondered half-way into their Euro 2024 last-16 game against Denmark if the gods were against them.

But as the storm clouds passed, so their fortunes changed and Germany deservedly ran out 2-0 winners in Dortmund to reach a first major tournament quarter-final in eight years.

That is a long time by the standards of a nation whose name became synonymous in football with serial winning.

The Germans' disappointing run has included first-round exits at the 2018 and 2022 World Cups and a last-16 elimination in the 2021 Euros.

Saturday's triumph has belatedly ignited Germany's fans who were notably downbeat about their prospects in the run-up to the tournament and had not created an atmosphere to match the famous summer party of 2006 when they hosted the World Cup.

That may change, as fans poured onto the streets after Saturday's games, tooting horns and waving flags.

The victory was vindication for Germany coach Julian Nagelsmann.

First and foremost of his decision to stick with Kai Havertz alone up front and keep big Niclas Fuellkrug on the bench despite his two goals at the tournament from substitute appearances.

Havertz, goalless in the group stage, looked the part, tormenting Denmark with his runs and clever movement and netting the first goal from the penalty spot with a pinpoint shot beyond goalkeeper Kasper Schmeichel's reach.

His fortune contrasted with Rasmus Hojlund at the other end, who looked shorn of confidence as his goal drought with Denmark continued to eight games.

Germany are starting to look very good at the Euros where in the group stage they humiliated Scotland, controlled Hungary and showed fighting spirit against a strong Swiss team.

Versus the Danes, they had an impressive 55 per cent possession, 58 attacks and 14 attempts on goal, Jamal Musiala curling home their second and match-winning goal.

They had the video assistant referee (VAR) to thank, however, for two game-changing decisions that will be a recurring nightmare for Danish defender Joachim Andersen.

He thought he had scored, only to see it chalked off for a narrow offside by a teammate, then shortly afterwards committed the handball for Germany's penalty.

“This is not how football is supposed to be,” Denmark coach Kasper Hjulmand said after the match.

“It was an even game. It was the referee who decided the game for us,” Andersen said.